K WAVENUMBER K WAVENUMBER

(b)

N I N

u I u

w w

(.() 104 ^(.() 104

N � u N � u

103

WINTER

1964 103

�

a

;;:..�

�

� �

\....-./

€ �

;;:..�

L.,.o�;-�-'

�v �--""

.,V v

vy

v

�

r7

^{'-- --1-- ---}

--1010

, /

/ /

�ol

v

)</ ^v

1;1' ./

109

v

/

/ /

length[2n/k(m)]

Fig. 5.3. Numerical result as calculated by formula (4.52). The ordinate indicates the ratio of the eddy viscosity coefficient to the molecular viscosity coefficient of the air. The abscissa indicates the characteristic length of a phenomenon, 2 n

I k.

We exa1nine the dependence of the eddy viscosity coefficient for Cheju Island scale on the parameter J1 of the power spectrum (see Fig. 5.4). The ordinate indicates the ratio of the eddy viscosity coefficient to the molecular viscosity coefficient of the air. The abscissa indicates the wave number at which the power spectrum is maximum, which is normalized by kmin· Although

...JJ.l

changes five orders, the ratio changes by less

than two orders at most. Hence the enormousness of the eddy viscosity coefficient is almost independent of

J.l.

We also examine the dependence of Veddy(k) on the truncation wave nmnber kmax·

Figure 5.5 is the dependence of the integrated value in (4.52), l(kmax)- /(k), on kmax• for Cheju Island scale, whose wave number is k =ke-D = 2 n

I (30

_km). The truncation wave number used in the previous estimation, kmax = I as kmin• corresponds to about

102

ke-D· The integrated values for kmax > 5 ke-D do not vary in significant figures of third­

order. Thus we conclude that the eddy viscosity coefficient Veddy(k) is dominantly

contributed by the modes whose wave numbers are almost equal to k in which we are interested and that the enormousness of the eddy viscosity coefficient is independent of the assumed parameters, kmax and

J.l.

In these respect, the enormousness of the eddy viscosity is independent of the shape of the spectrum.

... ... _�

... �

'"

5xl08

�

a

>�

...

� ...

"'

'\

'�

_�

�

⁸

2. 10

�

"\.. "

�

�

�

\...-/

' 1\.

' \.

� 108

\:3

I\

>� ^\ \

'

5xl07 ^\ _\

\ i\�

Fig. 5.4. Dependence of the eddy viscosity coefficient on the parameter J.1 of the power spectrum, for Cheju Island scale. The ordinate indicates the ratio of the eddy viscosity coefficient to the molecular viscosity coefficient of the air. The abscissa indicates the wave number normalized by kmfn, at which the power spectrum is maximum.

8

4

kmax J kC-D

Fig. 5.5. Dependence of the integrated value in (4.52) on the truncation wave number kmax for Cheju Island scale, whose wave number corresponds to ke-D ⁼ 2 n I (30 ktn). The ordinate indicates the integrated value in (4.52). The abscissa indicates the truncation wave number kmax divided by ke-D·

Equation (4.52) implies that eddies which have Cheju Island scale recognize the extent of earth, because it includes the scale of don1ain L. What is physics of that ? As we consider the properties of fluid confined within the earth as a basin of a finite size of L2, eddies of Cheju Island scale are regarded as phenomena in the basin. Therefore the earth and the eddies correspond to a heat bath and sub-systems contact with it in the statistical-thermodynatnics, respectively. Because the properties of the sub-systems depend on those of the heat bath, it stands to reason why the eddies recognize the extent of earth, i.e., the formula for the eddy viscosity coefficient ( 4.52) includes the size of domain L. For conserved systems, dynatnical properties of the systems are completely specified by initial conditions, by which conserved quantities are also determined.

However, properties of the systems in statistical mechanical theory are specified not by individual initial conditions but by conserved quantities. Since the scale of domain L and the parameters B and J1 determine the total energy and enstrophy, i.e., conserved quantities, it is also reasonable that these quantities are included in the formula for the eddy viscosity coefficient.

We can rewrite (4.52) without including L. By using (4.30), equation (4.52) bec01nes

where

= v l(k)

'

V = ' j

^Z+J.lE

'

\

/V

^{kmw?- kmin2}

\

l(k) = ^{4 J1 k4} 1(1

{

^{l(kmax) - /(k)}

} ·/

(5.2)

(5.3)

The former part of

(5.2),

V, is a constant that is determined by properties of total system, and has the dimension of velocity. The latter part,

l(k),

is a function of the wave number k, and has the dimension of length. Since the molecular viscosity coefficient

Vnzol

is expressed in terms of the product of the mean free path l of particles and the

mean value of Ouctuating velocity u of them, i.e.,

Vmol

� u I, the function l(k) con·esponds to the mean free path of vortices, or the mixing length. The form of

l(k)

_is

the same as that of the curve in Fig. 5.3, i.e.,

l(k)

�

k

-I (see Fig.

5.6).

For air at atmospheric pressure ( 1000 hPa) and room temperature (300°K) one has approxitnately u � 102 ms-1 and l � I 0 -7 m, hence

Vmol

�

1

_0-5 m2s-1 (Rei f,

1965).

On the other hand,

for the selected parameter values in this work, we have V ⁼

1.5

ms-1. We see by Fig.

5.6

that the enormousness of the eddy viscosity in comparison to the molecular viscosity is due to that of the function l(k), whereas l(k) is the function of the length scale of interest. Thus we conclude that the enonnousness of the eddy viscosity coefficient comes from that of the length scale of interest.

We consider what velocity V is. By using (4.30) again, we have

{E(z IE + p)

V ⁼

'V [k:'al

_�

_kmin

₂

₎

�---E

=

( ^kma.t

^{2 -}

^kmin

²

)

E

=

log

( ^kmax

²

^{+ J.1} )

kmin

2

+

^J.1

(5.4)

For the ens trophy equipartition state 1, that corresponds to f.1 ---7 0 (Kraichnan,

1975),

_we

have

1 In the ens trophy equipartition state, we have Z IE=

( ^kma}

⁺

^kmi/ )

^{I 2.}

(5.5)

If the truncation wave number kmax ⁼ I Qn kmin' where n is less than or equal to 7 at most2, we have

V -7 ^A

_{- V} /

_{4 n log} ^Uo2 ₁₀ ^{� 0.12 Uo. (5.6)}

Thus V is approxitnately the horizontal velocity scale. For the energy equipartition state3, that corresponds to J.l-7 f.1oo = 0.138033 for the selected parameter values of L, kmax and Uo in the previous estimation (Kraichnan, 1975), we obtain

V-79.5U0, (5.7)

i.e., Vis approximately the horizontal velocity scale or the characteristic speed of wind, again. In consequence, V approximately corresponds to the horizontal velocity scale, for states that all modes are significantly excited, those correspond to be the case of 0 < J.1 <

f.1oo (Kraichnan, 1975).

2 For the terrestrial condition, n = 7 corresponds to the wave length to be approximately 2.3 m.

If we are interested in phenomena whose wave lengths are less than that, the assumed conditions in this study, two-dimensional and inviscid fluid, may be violated, i.e., three-dimensionality and the molecular viscosity QlUSt be considered.

3 In this state, we have Z IE=

(

kmru? - kmin 2

)

^I

{

^{2 log}

⁽

kmrv: I kmin

) }·

� s

�

� �

�

�

500000.

105

50000.

104

5000.

/ L'

v /

/ v

1--^-

--_,.v

^�

./

-

-��

17 '

v

^v

��

v

� �

/ .£

v L

/., /

length [2n/k (m)]

� ...

�v

Fi�. ^5.6. Functional form of l(k). The ordinate indicates the value of l(k). The abscissa indicates the characteristic length, ^{2 n} I k.

§6. Concluding Remarks

Applying Mori's theory (Mori, 1965a) to the spectral form of the vorticity equation for the 2-D inviscid barotropic fluid, we derived the formula for the eddy viscosity coefficient. The eddy viscosity coefficient thus obtained depended on the length scale of a phen01nenon of interest. For Cheju Island scale, it was theoretically derived that the eddy viscosity coefficient was eight orders of magnitude larger than that of the molecular viscosity coefficient of the air, that was of the satne order as that of the eddy viscosity coefficient obtained by applying the Reynolds' law of sitnilarity to the atmospheric Karman vortices generated on the leeward of Cheju Island. Moreover, the enormousness of the eddy viscosity coefficient was independent of hypothesized spectrum shape and it was theoretically derived that the reason of the enormousness was due to that of the length scale of interest. This is the first study for theoretical elucidation of the enormousness of the atmospheric eddy viscosity coefficient from the fundamental equation of the fluid dynamics.

We comment on the application of the present discussion to atmospheric phenomena. We assumed that the system is of two-dimension and has the specific wind speed to be 10 ms-1• Although large scale atmospheric motions are actually three dimensional phenomena, two-dimensionality is approximately satisfied well. For example, the characteristic length of the synoptic scale disturbances that correspond to the high or low atmospheric pressure systems is about 3000 ktn in the horizontal and about 15 km in the vertical. Moreover, for atmospheric Karman vortices generated on the leeward of Cheju Island, the two-dimensionality is a good approximation. In general, the atmospheric temperature decreases monotonically with height from the ground to the tropopause that locates at an altitude of about 15 ktn. However a layer in which the temperature increases with height is locally observed. This layer is called an inversion layer. The atmospheric Karman vortices appear on the leeward of Cheju Island when the wind speed is more than 10 knots ( ^� 5 ms-1) and a well defined inversion layer exists at about 1 km altitude (Tsuchiya, 1969). Then the vertical motion

of the atmosphere is suppressed at this altitude. As seen from meteorological satellite photographs [e.g., see Tsuchiya, 1969], the horizontal scale of the atmospheric Karman vortices is 30 ^� 50 km. Thus the two-dimensionality is a good approxitnation in this case. In addition, it is seen that the specific wind speed assumed in this study is reasonable.

It is known that the eddy viscosity coefficient of the atmosphere has different value in both horizontal and vertical directions. The vertical eddy viscosity coefficient is 1 ^� 10 m2s-I, which varies with height, for the mesosphere, thus 5 ^� 6 order larger than the molecular viscosity of the air. That arises from the unstable breakdown of tides and gravity waves whose vertical wavelength is about 10 km (Lindzen, 1981 ).

Although our theory is the one for horizontal system, we can infer by the theory that the smallness of the vertical eddy viscosity coefficient in comparison to the horizontal one comes from the stnallness of vertical scale of atmospheric phenomena. This problem for three dimensional inviscid barotropic fluid remains to be discussed.

In this study, we did not consider effects of the terrestrial rotation or the beta effect. For a sufficiently large value of {3, a two-dimensional flow on a doubly-periodic beta-plane cannot be ergodic on the phase space of a constant energy and enstrophy (Shepherd, 1987). In consequence, the distribution function of the fluctuation of vorticity used in this study is invalid for such case. Since a distribution function of fluctuation of the vorticity including the beta-effect has not been known yet and the beta-effect violates the isotropy of the system, it is very hard to study the eddy viscosity for the system including the beta-effect, theoretically. This problem also retnains to be investigated.

Some parts of this study have been reported in the Progress of Theoretical Physics and the others (lwayama and Okamoto, 1993a; b; c).

Acknowledgments

I would like to thank Professor Hisao Okamoto of Kochi University and Professor Saburou Miyahara of Kyushu University for their eager guidance and continuous encouragement throughout my doctoral course, without which I would not have finished the work. Thanks are also extended to Professor Osamu Morita and Professor Tosihiko Hirooka of Kyushu University and Professor Masaaki Takahashi of University of Tokyo for their valuable cornments and encouragement. Discussions with Prof. 0. Morita roused my interests in the eddy viscosity coefficient. I wish to thank Dr. Yasumasa Ookouchi of Yatsushiro National College of Technology, Dr. Hideharu Akiyoshi of Fukuoka University, Ms. Yoshiko Sato, Mr. Satoru Takeuchi and all the other members of our dynamic n1eteorology group at Kyushu University and Mr. Atsusi Mori of University of Tokyo for their encouragement and Mr. E. M. P. Ekanayake for his work to clarify the n1anuscript. Dr. Yoshi-Yuki Hayashi and all the other participants of colloquium on this study at University of Tokyo provided me with useful information for writing the thesis. Mr. A. Mori provided me the opportunity for presentation at the colloquium. I am also grateful to Emeritus Professor Kanzaburou Gamba of University of Tokyo for his interest from the first stage of this study and valuable comments. His interest to this study gave the most encouragement to me.

Professor Michiya Uryu of Kyushu University also considered applications of the fluctuation-dissipation theorem to meteorological systems. Unfortunately, he went to final rest on August 2 I, 1990. I am very disappointed because I can not discuss this study with him. I would like to dedicate this volume to his soul.

Finally, sincere gratitude is extended to Professor Hiroshi Matsumura of Tokai University for his continuous guidance and encouragement from my student days in Tokai University until now. He firstly noticed the existence of Mori's theory to me.

References

Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press, 615pp.

Bell, T. L., 1980: Climate sensitivity from fluctuation dissipation: some simple tnodel tests. J. Atmos. Sci., 37, 1700-1707.

Charney, J. G., 1971: Geostrophic turbulence. J. Atmos. Sci., 28, l 087-1095.

Gambo, K., 1982: Vorticity equation of transient ultra-long waves in middle latitudes in winter regarded as Langevin's equation in Brownian motion. J.

Meteor. Soc. Japan, 60, 206-214.

Heisenberg, W., 1948a: Zur statistichen theorie der turbulenz. Z. Phys., 124, 628-657.

, 1948b: On the theory of statistical and isotropic turbulence. Proc. Roy. Soc.

London A, 195, 402-406.

Herring, J. R., and R. H. Kraichnan, 1972: Comparison of some approximations for isotropic turbulence. Statistical Models and Turbulence, M. Rosenblatt and C. Van. Atta, Eds. Springer-Verlag, 148-194.

Holton, J. R., 1979: An Introduction to Dynamic Meteorology, 2nd ed. Academic Press, 391 pp.

Iwayama, T. and H. Okamoto, 1993a: Fluctuation-dissipation theorem and eddy viscosity coefficient in two-di1nensional Rossby wave. The Technical Report of Research Institute for Mathematical Science (Kyoto University), No. 830, 157-166, (in Japanese).

, 1993b: Generalized Langevin equation for two-dimensional inviscid

barotropic vorticity: fluctuation-dissipation theorem and eddy viscosity. Prog.

Theor. Phys., 90, 343-351.

, 1993c: Theory of eddy viscosity coefficient for two-dimensional inviscid barotropic fluid. Prog. Theor. Phys., 90, No. 6 (December), (in press).

Kells, L. C. and S. A. Orszag, 1978: Randomness of low-order models of two­

dimensional inviscid dynamics. Phys. Fluids, 21, 162-168.

Kraichnan, R. H., 1959: The structure of isotropic turbulence at very high Reynolds number. 1. Fluid Mech., 5, 497-543.

, 1967: Inertial ranges in two-dimensional turbulence. Phys. Fluids, 10^, 1417-1423.

, 1970: Convergents to turbulence functions . .!. Fluid Mech., 41, 189-217.

, 1971 a: An almost-Markovian Galilean-invariant turbulence tnodel..!. Fluid Mech., 47, 513-524.

, 1971 b: Inertial-range transfer in two-and three-dimensional turbulence. 1.

Fluid Mech., 47, 525-535.

, I 975: Statistical dynamics of two-dimensional flow. 1. Fluid Mech., 67, 155-175.

, 1976: Eddy viscosity in two and three ditnensions. 1. Atmos. Sci., 33, 1521-1536.

Kubo, R., 1957: Statistical mechanical theory of irreversible processes. I: general theory and simple application to magnetic and conduction problems . .!. Phys.

Soc. Japan. 12, 570-586.

Kubo, R., M. Toda and N. Hashitsume, 1985: Statistical Physics II.

Nonequilibrium Statistical Mechanics. Springer-Verlag, 279pp.

Landau, L. D. and E. M. Lifshitz, 1976: Mechanics, 3rd ed. Pergamon Press, 169pp.

, 1987: Fluid Mechanics, 2nd ed. Pergamon Press, 539pp.

Leith, C. E., 1971: At1nospheric predictability and two-dimensional turbulence. 1.

Atmos. Sci., 28, 145-161.

Lifshitz, E. M. and L. P. Pitaevskii, 1981: Physical Kinetics. Pergamon Press, 452pp.

Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown. 1. Geophys. Res., 86, 9707-9714.

, 1990: Dynamics in Atmospheric Physics. Cambridge University Press, 310pp.

Lovesey, S. W., 1980: Condensed Matter Physics: Dynamic Correlations.

Benjamin, 191 pp.

, 1986: Condensed Matter Physics: Dynamic Correlations, 2nd ed. Benjamin, 370pp.

Mori, H., 1965a: Transport, collective motion and Brownian motion. Prog. Theor.

Phys., 33, 423-455.

, 1965b: A continued-fraction representation of time-correlation function.

Prog. Theor. Phys., 34, 399-416.

Prandtl, L., 1925: Bericht Uber untersuchungen zur ausgebildeten turbulenz. Zs.

angew. Math. Mech., 5, 136-139.

Reif, F., 1965: Fundamentals of Statistical and Thermal Physics. McGraw-Hill Book Co., 651 pp.

Shepherd, T. G., 1987: Non-ergodicity of in viscid two-dimensional flow on a beta plane and on the surface of a rotating sphere. f. Fluid Mech., 184, 289-302.

Stanisic, M. M., 1988: The Mathematical Theory of Turbulence, 2nd ed. Springer­

Verlag, 501 pp.

Tsuchiya, K., 1969: The clouds with the shape of Karman vortex street in the wake of Cheju Island, Korea. f. Meteor. Soc. Japan, 47, 457-465.

Appendix A. Explicit Form of R(k) and R

⁽l)

(k)

By calculating the inner products, which are listed in Appendix B, we have R(k) =

fi I

D(k, p) s(p) s(k-p)

p

=

(1- ?) I

D(k, p) s(P) s(k-p)

p

f (

^{s(P) s(k -}p ), s(k) ^*

) \

=

I

^{ock, p)}

\

^{s(p) sck -p)-}

(

*

)

^s(k)

/

p s(k), s(k)

=

I

D(k, p) ((p) ((k -P).

p

Next we calculate R(1)(k). We readily see that

(A.l)

=

(i I

^{D(k', q) s(q) s(k'-q) a}

I I

^\ p ock, p) s(p) sck- p) \1

k', q asck')

=

(2 {I

ocp, q) s(q) s(P- q) D(k, p) s(k-p)

p,q

+

I

^ock-p, q) s(q) s(k-p-q) ock, p) s(p)1\. cA.2)

p,q

By replacing k -p by p in the second term of the last expression of (A.2) and noticing that D(k, k-p) = D(k, p), we obtain

cA.2) = 2

fi I

ock, p) ocp, q) sck-p) scp-q) s(q).

p,q

Moreover, referring (B.4), we have

? I

ock, p) ocp, q) sck-p) scp- q) s(q)

p,q

- �

(

^{((k-p) ((p-}

^q)

^{((q), ((k) *}

)

- L.J D(k, p) D(p, q)

(

^*

)

^s(k)

p, q s(k), s(k)

(A.3)

and finally obtain

= I

_p

D(k, p) (D(p, k)

⁺

D(p, p- k)} ( ^{((k - p), ((k - p)}

^*

) ^((k)

=

₂

I

_p

D(k, p) D(p, k) ( ^{((k- p), ((k - p)}

^*

) ^((k)

^(A.4)

R

(l)(k)=

₂

I ock. p) ocp,

q)

sck - p) s-cp- q) seq)

p,q

(A.5) -4

I

_p

D(k, p) D(p, k) ( ^{((k - p), ((k - p)} *) ^((k).

Appendix B. List of the Inner Products

The inner products or the ensemble averages are given by

(4.14)

_{with the}

disttibution function

( 4.28).

It is readily seen that

form+ ⁿ being odd numbers, where ^{m =} km .

When m + ⁿ

= 4,

( Sol SC2l SOl. Sc 4 l * )

= ( Wl. SCil * ) ^{( Wl. SC3l }\- 2 81.4}

+

( Wl , Wl* ) ⁽ s(2), SC2l* ) o1.-3 02.4

+

( SCil, Wl * ) ⁽ SC2l, SC2l * ) 81.4 �.- 1

(B.l)

+

( w ) w ) , w / S(l) * )

^X

( ^{01, 2 03, - 4 01, - 3}

⁺

^{01, 3 �. - 4 01, -2}

( Wl SC2l. Wl * SC4 / )

+

81,-4 �. 3 81,-2 ) ^{, (B.2)}

= ( ^{Wl. Wl*} )( ^{sol. Wl*} ) o1.-2 0:3.-4

+

( ^{Wl. SCO *} )( SC2l, Wl * ) ^{81. 1 �. 4}

+

( ^{W ), w/)(} SC2l. SC2l * ) 81.4 �.

1

+

( S(l) W ), W l

•

SCI)* )

^x

( ^{81. 2} 0:3. 4 Ou

₊

81.

-1

�.

_ 4

81.

_

2

+

81,-4 �.-3 81,-2 ) ^{· (B.3)}

We show examples of (B.2) _and (B.3) for convenience of calculation in Appendix A and C:

( ((k - p) ((p- q) ((q), ((k) *)

=

( ((k - p), ((k - p) *)(s(k), ((k) *) �- p, q-p 8q, k

+

( ((k - p), ((k- p) *)( ({k), ((k) *) �

^_

p,

^_

q Dp

^_

q, k + ( ((p - q), ((p - q) *) ( ((k), ((k) *) _�

_

p, k Dp

_

q,

_

q

+

( ((k) ((k), ((k) * ((k) *)X ( �-p, p-q Dq,- k Dp-q,- q

( ((p) ((k- p), ({q). ((k - q) *)

+

� - p' q bp -q' -k 8p -q' -q

+

� -p, -k 8p

_

q, q �

_

p, q

_

p )

, ( B .4)

=

( ({p), ((p) *)( ((q), ((q) *) Dp,

P _

k Dq,q-k

+

( ((p), ({p) *)( ((k - p), ((k- p) *) Dp, q �

^_

p, k- q

+

( ((p), ((p) *)( ((k - p), ((k - p) *) Dp, k

^_

q 8k-p, q +(({p) ({p), ({p)• ((p)*)x(Dp, k-p 8q, k-q Dp, q

+

bp' - q � - p' q - k bp' k -p

+ 8p,q-k �-p,-q 8p, p-k)·

^(B.5)

We present an example of inner product form+ n = 6:

( ((k - p) ((p - q) ({q), ((k - p'/ ((p'- q'). ((q') *)

= 2

(S(k- pJ, ((k- Pl*)(S(qJ, ((k- p'J. ((p'- q'J. ((q'l*) ak

^_

p, q

^_

p

+ 2

( ((k - p), ((k- p) *)( ((p - q), ((k - p'). ((p'- q'). ((q') *) 8k- p,-q

+ ( S<k- p), S<k- Pl. )( ^S<P- q) S<q), S<P'- q')* S<q'l* ) ^Ok

^-

^{p. k- p'}

+

₂

( S<k- p), S<k- Pl. )( ^S<P- q) S<ql, S<k- p')* S<q'l* ) ok p, p'- q +

2

( ^{S<k -} p), S<k - p) * )( ^{S<P -} q) S<ql, S<k - p') * S<p'- q') * ) ^{ok- p, q}

+ ( ^{S<k - P l S<P -} q) S<q), S<k - p') * )( ^S< q), S<ql * ) ^{Op·- q·. q}

+ ( ^{S<k -} p) S<P - q), S<k - p') * S'(p' - q') * )( ^{S'(q), S<q) *} ) ^{Oq, q'}

+ ( ^{S'(k -} p) S<P- q), S<k - p') * S<q') * )( S'(q), S'(q) * ) ^{Oq, p'- q'}

+ ( ^{S<k -} p) S<q), S<k - p') * S<p'- q') * )( ^{S<P- q), S<P - q) *} ) ^{Op- q, q}

+ ( ^{S<k -} p) S'(q), S<k - p') * S(q') * )( ^{S<P- q), S<P- q) *} ) ^{Op- q, p'- q}

+ (( ^{S<k- Pl} ) ^3• ( ^{S<k- p)} T)

X

[ ^<\

^_

^{p, p}

^_

^{q 8p}

^_

q, q <\

_

p ', p'

_

q' 8p'

_

q ', q' <\ - p, k

-

p' + <\- p, p- q �- p,- q { ^{8p- q, p'-}

^k

^{8q, q'- p' 8p·- q', q'}

+ 8p - q, q'- p' 8q, p'- k �

-

p', q'

+ 8p - q, - q' 8q, p' - k � - p ', p' - q' }

+ <\- p, q � - p, q - p { ^{8q, p'- k 8p - q, q'- p' 8p·- q', q'}

+ 8q, q'- p' 8p- q, p'- k <\- p', q' +8q,

_

q' 8p

_

q, p'- k �- p', p'- q' }

+ 8p

_

q, q <\

_

p, q

_

p { ^{8q, p'}

^_

^{k <\}

^_

^{p, q'}

^_

^{p' 8p·}

^_

^{q', q'}

+ 8q, q'- p' <\

-

p, p'- k <\ - p', q'

+ 8q, _ ^{q · �}

^_

^{p, p}

^{· _}

^{k <\}

^_

^{p ·, p}

^{· _}

^{q ·} } _]. ^{(B. 6)}

Appendix C. Dete rmination of the Memory Function from the Fluctuation­

Dissipation Theorem

From (A. I) _and (8.5), _{we have}

(R(k), R(k)*)

⁼

L D(k, p) D(k, q) ( ^{((p) ((k - p), ((q) * ((k - q) *)}

p,q

= 2

L {D(k, p)jl ( ((p), ((p) * )( ((k- p), ((k - p) *).

p

Hence we obtain

_fO)- "' ²

(s<Pl. S<Pl. )(sck- Pl. S<k- Pl*)

rk

-2 LJ{D(k,p)}

( ^{* ) .}

p

s(k), s(k)

Now we define a positive quantity

p- 2

( ((p), ((p) * ) ( ^{((k- p), ((k - p) *)}

rk ⁼

2 { D(k, p)}

( s(k), s(k)

*

)

(C.

^I)

(C.2)

Since

(R

^(I

)(k), R(k)*)

includes only the ensemble average of odd numbers of Inodes, we have

(R (l) (k), R(k)*)

^{= 0}

and

Next we see that

= 4

L D(k, p) D(p, q) D(k, p') D(p', q')

p, q, p', q'

(C.3)

X

( c;-(k - p) c;-(p - q) c;-(q), c;-(k - p')

^•

c;-(p'- q') * c;-(q') *) -

¹⁶

^L [ D(k, p) D(p, k) D(k, p') D(p', k) (sck - p'), sck - p') *)

p, p'

+ 16

L [ D(k, p) D(p, k) D(k, p') D(p', k) ( c:-<k - p'), c:-<k - p') *)

p, p'

X

( c;-(k- p), c;-(k- p) *) ( c;-(k), c;-(k) *)].

^(C.4)

By making use of (B.4), the second term of (C.4) reduces to

- 16

L [ D(k, p) D(p, k) D(k, p') D(p'' k) ( c:-<k - p '), c:-<k - p ') *)

p, p'

(C.5)

Using (B.6), (B.2) and (B.3), and noticing that

D(k, k- p)

⁼

D(k, p),

the first term of (C.4) becomes

L

2 ykP Ypq

( S<k), S(k) *)

p,q

+ 8 _p,q

L {D(k, p - q) D(k - p

⁺

q, q)

⁺

D(k - q, p- q) D(k, q)}

X

D(k, p) D(p, q) ( c:-<k - p ), c:-<k - p) *) ( S<P - q), S<P - q) *) ( c:-< q), c:-< q) *)

+ 16

L [ D(k, p) D(p, k) D(k, p') D(p', k) ( c:-<k - p'), c:-<k - p ') *)

p, p'

X

(c:-(k- p), c;-(k- p)*)(c:-(k), c;-(k)*) l

+ 4

L

_p

[

⁴

(D(k, p) D(p, k- pl)Z (S<k- p) S<k- p), S<k- p). S<k- p/)

X

( ^S

⁽²

p - k), S<2 p - k) *)].

^(C.6)

Replacing

p - q

^by

q

in the first term in the curly bracket of the second term of (C.6), the second tetm of (C.6) reduces to

16

L D(k, q) D(k - q, p- q) D(k, p) D(p, q)

p,q

X

( ((k - p), ((k - p) *) ( ((p - q), ((p - q) *) ( ((q), ((q) *).

_(C.7)

Moreover, replacing

k - p

_by

p,

we obtain

(C.7) ⁼ 16

L D(k- p, q) D(k- q, p) D(k, p) D(k, q)

p, q

The third term of (C.6) is canceled by adding itself with (C.5). With the aid of (C.2), the last tenn of (C.6) is rewritten as

(sck- Pl sck- p), sck- Pl. sck- Pl*)(sckl, S<kl*)

� 4 y. P � k- P (C. 9)

L k P

( *)2

p

sck - p ), sck - p)

Therefore we have

�2)

=

L

2 ykP Ypq p,q

+ 16

L D(k- p, q) D(k- q, p) D(k, p) D(k, q)

p,q

(sck- P- q), S<k- P- q)*)(scp), S<Pl*)(scq), S<ql*)

X

-( -(-(k), ((k) *)

(sck- p) s(k- p), s(k- p)* s(k- p)*)

+

L

4 II p � k- p .:_ _______ _

p k p

( sck -

p

), sck - p) T

(C.l 0)

Appendix ^D. Integration in (4.52)

The integration for the angular variable in (4.52) is evaluated with the aid of the complex integral:

where

and

1 sin2e

(a-

^{cos e}

)

i

^{2 TC 2}

"' 0 de 2p k

(

^{b-cos e}

) (

^{c-cos e}

) (

^d

-

^{cos e}

)

"'�

_P

_l 1 _c

^B(z)dz,

\ I

i

(

^{z 2 - 1}

_Y (

^{z 2 -2}

a

^{z + 1}

)

²

�z) =

---�--�---�---2z2

(

^{z2-2bz+ I}

)(

^{z2-2cz+ I}

){

^{z2-2dz+ I}

r

(D.l)

(D.2)

(D.3)

Note that b � I, c > 1 and d > I, because p > 0, k > 0 and f.1 > 0. The variable z is the complex variable with magnitude of unity,

z = ei e. (D.4)

The last expression of

(D.l)

is integrated along a curve C which is the unit circle with center at the origin in the z plane.

The poles of integrand of (D. 1 ), B(z), are z

=

0, z

=

_b ±

Y

h2- I, z

=

c ± c2- 1 and z

= d

±

Y d2

- 1 . Among them, four poles locate within the unit circle C. These are

Z 1

=

^0,

Z

2

=

h-

�

b2- } ,

}

Z3=C-�, Z4=d - �.

The residues of

�z)

at these poles are

and

. .4p2+k2+J.1

Re"r '1 z

1] =

z (2 ^a- h- c-

d)=-

z ----2pk ^'

R _e"'z2 "r

J- _-

_l .(a-b)_{b- 2�- .P

V

^J2-k2

/

c

)

{b

-d) -

^l 2 J1 k ,

Next, the integration for the radial coordinate is evaluated as follows:

(D

. ₅₎

(D.6)

(D.7)

(D.8)

(D.9)

(D.lO)

(0.11)

(0.12)

- (

K2-2

J1)

log

[ _�(p)

+

X(P)]} ��max,

_(0.13)

(0.14)

Parameters _{Kt, K2} and _K3 and functions

�(p ), x(p )

^,

TJ(p)

^and

i}(jJ)

are defined in the text.

ドキュメント内 2次元非粘性順圧流体に関する渦粘性の理論 (ページ 57-87)